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is equal to one-third the product of its base by its altitude (Theo. VII).

Cor. 2. The solidity of any pyramid whatever, is equal to one-third the product of its base by its altitude; for, by dividing its base into triangles, and passing planes through the lines of division and the vertex, the pyramid can be divided into a number of triangular pyramids; and the sum of their solidities. will be equal to one-third the product of the sum of their bases by their common altitude.

THEOREM XII.

The convex surface of a cone is equal to half the product of the circumference of the base by the slant hight; and its solidity is equal to one-third the product of the area of the base by the altitude.

Let ABC be a cone, having a regular pyramid inscribed in it. If the number of sides of the polygon constituting the base of the pyramid be indefinitely increased, its perimeter will ultimately coinIcide with the circumference of the base of the cone. Then, the slant hight of the pyramid will be equal

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to the slant hight of the cone, and the convex surface and solidity of the pyramid to the convex surface and solidity of the cone. But the convex surface of the pyramid will be equal to half the product of the perimeter of the base by the slant hight (Theo. IX); and the solidity of the pyramid will be equal to one-third

the product of the area of the base by the altitude (Cor. 2, Theo. XI).

Therefore, the convex surfaces of a cone, etc.

Schol. In the same manner, it may be shown that the convex surface of a frustum of a cone is equal to the product of the slant hight into the circumference of a middle section between the two bases (Schol. 2, Theo. IX). And as this will hold true however small the upper base may be, it will hold true of the cone itself, which may be treated as a frustum whose upper base is nothing.

SEC. XVII.-THE SPHERE.

DEFINITIONS.

1. A SPHERE is a solid which may be described by the revolution of a semicircle around its diameter as a fixed axis.

The semi-circumference describes the convex surface. The center is the middle point of the axis. The radius is a straight line from the center to any point of the surface; and it is equal to the radius

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of the semicircle. A diameter is a double radius.

Schol. As the semicircle ADB revolves about the axis AB, a perpendicular, as DC, let fall on the axis from any point in the circumference, will describe a circle.

2. A GREAT CIRCLE on the sphere is one whose

plane passes through the center. Its radius is the same as the radius of the sphere. Its circumference is also called the circumference of the sphere. Any other circle on the sphere is called a SMALL CIRCLE.

3. A SEGMENT of a sphere is any portion, as ADEF, cut off from the solid by a plane passing through it.

4. A frustum of a cone is said to be inscribed in a sphere, when the circumferences of its bases lie in the surface of the sphere.

THEOREM XIII.

If a frustum of a cone be inscribed in a sphere, its convex surface will be equal to the altitude of the frustum multiplied by the circumference of a circle, whose radius is a perpendicular from the center of the sphere to the slant hight of the frustum.

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Let CD and GH be both perpendicular to the axis AB. Then, as the semicircle revolves about the axis, describing the sphere, the trapezoid DCGH will describe a frustum of a cone, which will be inscribed in the sphere. From the center K let fall the perpendicular KE, on the chord CG; then E will be the middle point of CG (Theo. XXV, B. I). Draw EF perpendicular to AB, and CI perpendicular to GH.

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Now, since the triangles GCI, KEF, have the sides of the one respectively perpendicular to the sides of the other, they are similar (Theo. VI, B. II).

Hence, GC CI :: KE : EF.

Multiplying an extreme and a mean equally (Def. 5, Sec. X),

GC CI 2 KEX3.14159: 2EFX3.14159. The first term of this proportion is the slant hight of the frustum; the second is its altitude; the third is the circumference of a circle whose radius is KE (Cor. 1, Theo. XV, B. II); the fourth is the circumference of a circle whose radius is EF. Therefore, multiplying extremes and means, we have the product of the slant hight into the circumference of a middle section of the frustum, equal to the product of its altitude into the circumference of a circle whose radius is the perpendicular from the center to the slant hight. But the former product is equal to the convex surface of the frustum (Schol., Theo. XII); consequently, the latter product is equal to the same. That is, if a frustum of a cone be inscribed, etc.

Schol. If the chord CG be one side of a regular inscribed polygon, it is evident that KE will be its apothegm (Cor. 2, Theo. XXVIII, B. I).

THEOREM XIV.

The surface of a sphere is equal to the product of its diameter by its circumference.

Let ABCDEF be a semicircle, having the half of a regular polygon inscribed in it. As the semicircle revolves about the axis AF, describing the sphere, each of the trapezoids GBCH, HCDI, etc., will describe a frustum of a cone, which

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will be inscribed in the sphere. The convex surface

of each of these frustums will be equal to its altitude multiplied by the circumference of a circle whose radius is the apothegm of the polygon (Schol., Theo. XIII). Therefore, the sum of their convex surfaces will be equal to the circumference of such a circle multiplied by the sum of the altitudes GH, HI, etc., that is, multiplied by AF, the diameter of the sphere. Now if the number of sides of the semi-polygon be indefinitely increased, its perimeter will ultimately coincide with the semi-circumference, and its apothegm with the radius of the sphere. Then, the sum of the convex surfaces of the frustums will be equal to the surface of the sphere, and the circumference of which the apothegm is radius will be the circumference of the sphere (Def. 2).

Therefore, the surface of a sphere is equal, etc.

Cor. 1. Since the circumference is equal to 3.14159 XD (Cor. 1, Theo. XV, B. II), it follows that the surface is equal to 3.14159×D2.

Cor. 2. The convex surface of any segment of a sphere, as that described by the arc ABC, is equal to its altitude AH multiplied by the circumference of the sphere.

THEOREM XV.

The solidity of a sphere is equal to its surface multiplied by one-sixth of its diameter.

The sphere may be considered as made up of infinitely small pyramids, whose bases together form the surface of the sphere, and whose common vertex is at the center. Now, the solidity of each of these pyramids will be equal (Cor. 2, Theo. XI) to the product

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